Stability conditions for the explicit integration of projection based nonlinear reduced-order and hyper reduced structural mechanics finite element models
For researchers and engineers using reduced-order models in structural mechanics, this work offers a theoretical guarantee and practical formulas for stable time integration, potentially speeding up simulations.
This paper provides a theoretical analysis showing that Galerkin projection-based reduced-order models in structural mechanics have stable time step sizes that are always larger than or equal to those of the full-order model, with mathematical expressions for computing the stable step size. The findings enable faster simulations by allowing larger time steps.
Projection-based nonlinear model order reduction methods can be used to reduce simulation times for the solution of many PDE-constrained problems. It has been observed in literature that such nonlinear reduced-order models (ROMs) based on Galerkin projection sometimes exhibit much larger stable time step sizes than their unreduced counterparts. This work provides a detailed theoretical analysis of this phenomenon for structural mechanics. We first show that many desirable system matrix properties are preserved by the Galerkin projection. Next, we prove that the eigenvalues of the linearized Galerkin reduced-order system separate the eigenvalues of the linearized original system. Assuming non-negative Rayleigh damping and a time integration using the popular central difference method, we further prove that the theoretical linear stability time step of the ROM is in fact always larger than or equal to the critical time step of its corresponding full-order model. We also give mathematical expressions for computing the stable time step size. Finally, we show that under certain conditions this increase in the stability time step even extends to some hyper-reduction methods. The findings can be used to compute numerical stability time step sizes for the integration of nonlinear ROMs in structural mechanics, and to speed up simulations by permitting the use of larger time steps.